论文标题
$ \ MATHCAL {P} \ MATHCAL {T} $ - 对称$-Gφ^4 $理论
$\mathcal{P}\mathcal{T}$-symmetric $-gφ^4$ theory
论文作者
论文摘要
具有潜在$ v（φ）= \ textstyle {\ frac {\ frac {1} {2}} m^2φ^2- \ textstyle {\ frac {\ frac {1} {4}}gφ^4 $（$ g> 0 $）的标量字段理论是一种源泉，但在非毛刺理论中， $ \ MATHCAL {P} \ MATHCAL {T} $ - 对称框架的定义很好，并且对于时空尺寸$ d = 1 $，它具有正面的真实能量谱。虽然文献中使用的方法并不容易推广到量子场理论，但在本文中，$ \ Mathcal {p} \ Mathcal {t} $ - 对称$-Gφ^4 $理论的路径集成表示形式显示为通用$ d $提供统一的配方。欧几里得分区功能之间的一个新的猜想关系$ z^{\ Mathcal {p} \ Mathcal {t}}}（g）$ （$λ> 0 $）提出了Hermitian理论：$ \ log z^{\ Mathcal {p} \ Mathcal {t}}}}（g）= \ textStyle {\ frac {\ frac {1} {2}} {2}} \ log z _ { 0^+）+\ textstyle {\ frac {1} {2}}} \ log z _ {\ rm herm}（ - g - g - {\ rm i} 0^+）$。这种关系确保了非Hermitian $ \ MATHCAL {P} \ MATHCAL {T} $ - 对称$-Gφ^4 $字段理论的真实能量谱。密切相关的关系在$ d = 0 $中严格有效。对于$ d = 1 $，使用$ z^{\ Mathcal {p} \ Mathcal {t}}}（g）$的半经典评估，通过比较地基能量$ e_0^{\ Mathcal {p} \ Mathcal {p} \ Mathcal {tecelt and can（g）（can）（can）（can）（can）（ $ e_ {0，\ rm herm}（-g \ pm {\ rm i} 0^+）$。
The scalar field theory with potential $V(φ)=\textstyle{\frac{1}{2}} m^2φ^2-\textstyle{\frac{1}{4}} gφ^4$ ($g>0$) is ill defined as a Hermitian theory but in a non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric framework it is well defined, and it has a positive real energy spectrum for the case of spacetime dimension $D=1$. While the methods used in the literature do not easily generalize to quantum field theory, in this paper the path-integral representation of a $\mathcal{P}\mathcal{T}$-symmetric $-gφ^4$ theory is shown to provide a unified formulation for general $D$. A new conjectural relation between the Euclidean partition functions $Z^{\mathcal{P}\mathcal{T}}(g)$ of the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric theory and $Z_{\rm Herm}(λ)$ of the $λφ^4$ ($λ>0$) Hermitian theory is proposed: $\log Z^{\mathcal{P}\mathcal{T}}(g)=\textstyle{\frac{1}{2}} \log Z_{\rm Herm}(-g+{\rm i} 0^+)+\textstyle{\frac{1}{2}}\log Z_{\rm Herm}(-g-{\rm i} 0^+)$. This relation ensures a real energy spectrum for the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric $-gφ^4$ field theory. A closely related relation is rigorously valid in $D=0$. For $D=1$, using a semiclassical evaluation of $Z^{\mathcal{P}\mathcal{T}}(g)$, this relation is verified by comparing the imaginary parts of the ground-state energy $E_0^{\mathcal{P}\mathcal{T}}(g)$ (before cancellation) and $E_{0,\rm Herm}(-g\pm {\rm i} 0^+)$.
